Abstract

We investigate the topological edge modes of surface plasmon polaritons (SPPs) in a non-Hermitian system composed of graphene pair arrays with alternating gain and loss. The topological edge modes emerge when two topologically distinct graphene arrays are connected. The gain and loss present in the system provide additional ways to control the propagation loss and field distributions of the topological edge modes. Moreover, the existence of the topological edge modes is related to the broken parity-time (PT) symmetry. We show the beam diffraction can be steered by tuning the chemical potential of graphene. Thanks to the strong confinement of SPPs, the topological edge modes can be squeezed into a lateral width of ~λ/70. We also show such modes can be realized in lossy graphene waveguides without gain. The study provides a promising approach to realizing robust light transport and optical switches on a deep-subwavelength scale.

© 2017 Optical Society of America

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References

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    [Crossref]
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  43. S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
    [Crossref]
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    [Crossref]

2017 (7)

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

F. Wang, Z. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Asymmetric plasmonic supermodes in nonlinear graphene multilayers,” Opt. Express 25(2), 1234–1241 (2017).
[Crossref] [PubMed]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
[Crossref]

2016 (10)

X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
[Crossref]

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

P. Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

X. Lin, R. Li, F. Gao, E. Li, X. Zhang, B. Zhang, and H. Chen, “Loss induced amplification of graphene plasmons,” Opt. Lett. 41(4), 681–684 (2016).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

H. Deng, X. Chen, N. C. Panoiu, and F. Ye, “Topological surface plasmons in superlattices with changing sign of the average permittivity,” Opt. Lett. 41(18), 4281–4284 (2016).
[Crossref] [PubMed]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

2015 (4)

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
[Crossref] [PubMed]

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23(17), 21585–21595 (2015).
[Crossref] [PubMed]

2014 (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

M. Ornigotti and A. Szameit, “Quasi-symmetry in passive photonic lattices,” J. Opt. 16(6), 065501 (2014).
[Crossref]

2013 (3)

Y. M. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
[Crossref]

X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
[Crossref]

H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38(11), 1912–1914 (2013).
[Crossref] [PubMed]

2012 (6)

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
[Crossref] [PubMed]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

2011 (3)

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[Crossref]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

2010 (2)

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

2008 (3)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

Alù, A.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Badioli, M.

G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
[Crossref] [PubMed]

Bao, Q.

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

Basov, D. N.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

Berini, P.

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010).
[Crossref]

Bernechea, M.

G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
[Crossref] [PubMed]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Bliokh, K. Y.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

Bolotin, K. I.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

Boubanga-Tombet, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

Castro Neto, A. H.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

Chan, C. T.

Chan, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

Chen, H.

X. Lin, R. Li, F. Gao, E. Li, X. Zhang, B. Zhang, and H. Chen, “Loss induced amplification of graphene plasmons,” Opt. Lett. 41(4), 681–684 (2016).
[Crossref] [PubMed]

X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
[Crossref]

Chen, J.

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

Chen, P. Y.

P. Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Chen, X.

Cheng, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
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G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
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F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
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C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
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G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
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D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
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G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
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K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
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K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
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G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
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G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
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C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
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D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

Li, E.

Li, H.

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

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C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
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Li, R.

Li, T.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
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T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
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C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
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G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
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X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
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X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
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Lumer, Y.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
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Luo, L.

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

Luo, X.

X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
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Makris, K. G.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

Martín-Moreno, L.

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
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McLeod, A. S.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
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V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

Moiseyev, N.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

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G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

Ni, Z.

X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
[Crossref]

Nikitin, A. Y.

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
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Nolte, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
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Nori, F.

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
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M. Ornigotti and A. Szameit, “Quasi-symmetry in passive photonic lattices,” J. Opt. 16(6), 065501 (2014).
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G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
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S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
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V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

Özyilmaz, B.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

Pan, Y.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Panoiu, N. C.

Plotnik, Y.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

Qin, C.

Qiu, C. W.

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

Qiu, T.

X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
[Crossref]

Rechtsman, M. C.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Rivera, N.

X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
[Crossref]

Rong, K.

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

Ryzhii, M.

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

Ryzhii, V.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

Satou, A.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

Schmalian, J.

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

Schomerus, H.

Segev, M.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

Shen, S.

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

Shi, W.

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

Sikes, K. J.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

Soljacic, M.

X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

Stormer, H. L.

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

Sun, C.

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

Szameit, A.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

M. Ornigotti and A. Szameit, “Quasi-symmetry in passive photonic lattices,” J. Opt. 16(6), 065501 (2014).
[Crossref]

Teng, J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Tian, J.

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

Tringides, M. C.

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

Wagner, M.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

Wang, B.

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
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F. Wang, Z. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Asymmetric plasmonic supermodes in nonlinear graphene multilayers,” Opt. Express 25(2), 1234–1241 (2017).
[Crossref] [PubMed]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Wang, F.

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

F. Wang, Z. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Asymmetric plasmonic supermodes in nonlinear graphene multilayers,” Opt. Express 25(2), 1234–1241 (2017).
[Crossref] [PubMed]

Wang, J.

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

Wang, K.

Wang, L.

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

L. Ge, L. Wang, M. Xiao, W. Wen, C. T. Chan, and D. Han, “Topological edge modes in multilayer graphene systems,” Opt. Express 23(17), 21585–21595 (2015).
[Crossref] [PubMed]

Wang, Q.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Wang, Y.

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

Wang, Z.

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

F. Wang, Z. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Asymmetric plasmonic supermodes in nonlinear graphene multilayers,” Opt. Express 25(2), 1234–1241 (2017).
[Crossref] [PubMed]

Watanabe, T.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

Weimann, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

Wen, W.

Xia, F.

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

Xiao, M.

Yao, K.

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

Ye, F.

Yuan, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Zhai, C.

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

Zhang, B.

Zhang, J.

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

Zhang, Q.

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

Zhang, X.

X. Lin, R. Li, F. Gao, E. Li, X. Zhang, B. Zhang, and H. Chen, “Loss induced amplification of graphene plasmons,” Opt. Lett. 41(4), 681–684 (2016).
[Crossref] [PubMed]

Y. M. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
[Crossref]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Zhao, D.

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Zhao, H.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
[Crossref] [PubMed]

Zhu, S.

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Zhu, X.

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

ACS Nano (2)

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011).
[Crossref] [PubMed]

Adv. Mater. Technol. (1)

F. Gan, C. Sun, Y. Wang, H. Li, Q. Gong, and J. Chen, “Multimode metallic double-strip waveguides for polarization manipulation,” Adv. Mater. Technol. 2(4), 1600248 (2017).
[Crossref]

Appl. Phys. Lett. (2)

Y. M. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
[Crossref]

H. Da and C. W. Qiu, “Graphene-based photonic crystal to steer giant Faraday rotation,” Appl. Phys. Lett. 100(24), 241106 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

F. Wang, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Rabi oscillations of plasmonic supermodes in graphene multilayer arrays,” IEEE J. Sel. Top. Quantum Electron. 23(1), 125 (2017).
[Crossref]

J. Appl. Phys. (1)

V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. (1)

M. Ornigotti and A. Szameit, “Quasi-symmetry in passive photonic lattices,” J. Opt. 16(6), 065501 (2014).
[Crossref]

Laser Photonics Rev. (1)

Q. Cheng, Y. Pan, Q. Wang, T. Li, and S. Zhu, “Topologically protected interface mode in plasmonic waveguide arrays,” Laser Photonics Rev. 9(4), 392–398 (2015).
[Crossref]

Mater. Sci. Eng. Rep. (1)

X. Luo, T. Qiu, W. Lu, and Z. Ni, “Plasmons in graphene: recent progress and applications,” Mater. Sci. Eng. Rep. 74(11), 351–376 (2013).
[Crossref]

Nanotechnology (1)

C. Sun, K. Rong, Y. Wang, H. Li, Q. Gong, and J. Chen, “Plasmonic ridge waveguides with deep-subwavelength outside-field confinements,” Nanotechnology 27(6), 065501 (2016).
[Crossref] [PubMed]

Nat. Mater. (1)

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2016).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, F. Gatti, and F. H. Koppens, “Hybrid graphene-quantum dot phototransistors with ultrahigh gain,” Nat. Nanotechnol. 7(6), 363–368 (2012).
[Crossref] [PubMed]

Nat. Photonics (4)

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. McLeod, M. K. Liu, F. Keilmann, B. Özyilmaz, A. H. Castro Neto, J. Hone, M. M. Fogler, and D. N. Basov, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
[Crossref]

New J. Phys. (1)

X. Lin, N. Rivera, J. J. López, I. Kaminer, H. Chen, and M. Soljačić, “Tailoring the energy distribution and loss of 2D plasmons,” New J. Phys. 18(10), 105007 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

D. Zhao, Z. Wang, H. Long, K. Wang, B. Wang, and P. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Phys. Rev. A (1)

C. Zhai, X. Zhu, P. Lan, F. Wang, L. He, W. Shi, Y. Li, M. Li, Q. Zhang, and P. Lu, “Diffractive molecular-orbital tomography,” Phys. Rev. A 95(3), 033420 (2017).
[Crossref]

Phys. Rev. Appl. (1)

P. Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

Phys. Rev. B (2)

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85(3), 035443 (2012).
[Crossref]

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[Crossref]

Phys. Rev. Lett. (6)

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degeneracies, and topological numbers in Non-Hermitian systems,” Phys. Rev. Lett. 118(4), 040401 (2017).
[Crossref] [PubMed]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108(16), 167401 (2012).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Sci. Rep. (2)

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in Non-Hermitian optical materials,” Sci. Rep. 5(1), 17022 (2015).
[Crossref] [PubMed]

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. Rep. 5(1), 12423 (2015).
[Crossref] [PubMed]

Solid State Commun. (1)

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

Other (4)

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” https://arxiv.org/abs/1702.02553 (2017).

L. Li, X. Zhu, P. Lan, L. He, and P. Lu, “Photon channel perspective on high harmonic generation,” https://arxiv.org/abs/1702.04084 (2017).

F. Bleckmann, S. Linden, and A. Alberti, “Spectral imaging of topological edge states in plasmonic waveguide arrays,” https://arxiv.org/abs/1612.01850 (2016).

Z. Wang, B. Wang, H. Long, K. Wang, and P. Lu, “Surface plasmonic lattice solitons in semi-infinite graphene sheet arrays,” https://arxiv.org/abs/1611.06637 (2016).

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Figures (9)

Fig. 1
Fig. 1 Schematic of graphene pair arrays. The structure is periodic along x direction. In each unit cell, there are two graphene sheets with alternating gain (red) and loss (blue).
Fig. 2
Fig. 2 The dispersion relation and mode profile of SPPs in the PT-symmetric graphene pair arrays. (a) and (b) Real (blue line) and imaginary (red line) parts of the propagation constants for (a) d2 = 120 nm and (b) d2 = 80 nm. In all cases, d1 = 80 nm. (c)-(f) The field distributions (Hy) at φ = 0 and φ = π correspond to (a). The real part of the modes is illustrated in blue and the imaginary part in red.
Fig. 3
Fig. 3 The dispersion relation and the topological invariant. (a) Real and (b) imaginary part of dispersion relation as the interlayer spacing is varying. (c) Real and (d) imaginary parts of winding numbers as a function of d2. The blue dots (red line) represent the lower (upper) band. In all cases, d1 = 80 nm.
Fig. 4
Fig. 4 Two arrays connect through (a) a short-short and (b) a long-long defects. The center graphene sheet is with gain. The green arrows denote the unit cell. (c) and (d) are the eigenvalues of supermode corresponding to (a) and (b), respectively. (e) and (f) show the mode profiles (|H|) of the topological edge modes for (a) and (b), respectively. Two trivial defect modes (purple and green) are also shown in Fig. 4(e).
Fig. 5
Fig. 5 The band diagram and the mode profile for a different gain and loss distribution. The center graphene sheet now is lossless. (a) and (b) show the structures with short-short and long-long defects, respectively. (c) and (d) are the eigenvalues of all supermodes. (e) and (f) show the mode profiles (|(H)|) of the topological edge modes and two trivial defect modes (purple and green) at the interface.
Fig. 6
Fig. 6 The real part of eigenvalues for different waveguide spacing d2. Here d2 refers to dL 1 = dR 2. The spacing dL 2 and dR 1 are fixed at 80 nm. (a) is for the structure shown in Fig. 4 and (b) for Fig. 5. The blue (red) dots represent the eigenvalues for the short-short (long-long) defects. When 70nm < d2 < 96nm, the PT symmetry is broken (green regions).
Fig. 7
Fig. 7 The light evolutions for a single-waveguide excitation corresponding to the structures shown in Fig. 5. (a) Two graphene arrays connected by a short-short defect. (b) The arrays with equidistance. (c) The arrays connected by a long-long defect.
Fig. 8
Fig. 8 The influences of chemical potential on the beam propagations. (a) The regions of broken PT symmetry as a function of chemical potential for the infinite graphene arrays. The intra-layer spacing is fixed at d1 = 80 nm. (b) and (c) are the light evolutions for μc = 0.16 eV and μc = 0.137 eV, respectively. The structures are the same as that shown in Fig. 7(c).
Fig. 9
Fig. 9 The topological edge modes in passive PT-symmetric graphene multilayers. (a) and (b) are the spectra of the supermodes for the structure shown in Figs. 4(b) and 5(b), respectively. The “lossy” waveguide now is with τ = 0.5 ps, the “gain” with τ = 1 ps, and the lossless with τ = 0.67 ps.

Equations (6)

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cos(φ)= ξ 1 ξ 2 κ 2 2 sinh(κ d 1 )sinh(κ d 2 ) ( ξ 1 + ξ 2 )κ 2 sinh(κd)+cosh(κd).
H=( β 0 iγ/2 c 1 + c 2 e iφ c 1 + c 2 e iφ β 0 +iγ/2 ),
β ± = β 0 ± c 1 2 + c 2 2 +2 c 1 c 2 cosφ γ 2 /4 .
W h = i π π/d π/d d k x Ψ m ( k x ) | k x | Ψ m ( k x ) ,
β A 1 =( β 0 i γ 2 ) A 1 + c 1 B 1 + c 2 B 2 , β B 1 =( β 0 +i γ 2 ) B 1 + c 2 A 0 + c 1 A 1 , β A 0 =( β 0 i γ 2 ) A 0 + c 2 B 1 + c 2 B 1 , β B 1 =( β 0 +i γ 2 ) B 1 + c 2 A 0 + c 1 A 1 , β A 1 =( β 0 i γ 2 ) A 1 + c 1 B 1 + c 2 B 2 ,
A n =a σ |n| , B n ={ b σ |n| ,n<0 c σ |n| ,n>0 ,

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